Geometric meaning of vanishing of higher cohomology of quasi-coherent modules over affine schemes

Question

One of the basic vanishing results about quasicoherent (sheaves of) modules over affine schemes is that their non-zero cohomology vanishes.

My only geometric intuition for sheaf cohomology is via Čech cohomology, where $H^n$ measures how many sections are gained as we go local with increasing compatibility conditions.

My only intuition for quasicoherent modules over affine schemes is their equivalence to modules over the corresponding ring.

Is the vanishing theorem something that should be seen as tautological? What is its geometric content?

Answer 1

That the Čech cohomology of a quasi-coherent module on an affine scheme vanishes corresponds to the observation that for some commutative ring $A$, some finite sequence of elements $(f_i)_{i \in I}$ of $A$ generating $A$ as an ideal, and some $A$-module $M$, the following sequence is exact: $$(*) ~~~~ 0 \to M \to \prod_{i \in I} M[f_i^{-1}] \to \prod_{i,j \in I} M[f_i^{-1},f_j^{-1}] \to \prod_{i,j,k \in I} M[f_i^{-1},f_j^{-1},f_k^{-1}] \to \dotsc$$ Here, the boundary maps are given as alternating sums of localization maps (which we may write as if they were inclusions, if we are careful). For example,$$\prod_{i,j \in I} M[f_i^{-1},f_j^{-1}] \to \prod_{i,j,k \in I} M[f_i^{-1},f_j^{-1},f_k^{-1}]$$ maps $(m_{ij})_{i,j}$ to $(m_{jk}-m_{ik}+m_{ij})_{i,j,k}$.

I would speculate that this is a fact from commutative algebra which cannot be proven or even formulated purely geometrically. The proof is a direct calculation; one explicitly constructs preimages of elements in the kernels of the boundaries.

Notice that the beginning of the exact sequence $(*)$

$$0 \to M \to \prod_{i} M[f_i^{-1}] \to \prod_{i,j} M[f_i^{-1},f_j^{-1}]$$

is already common from the construction of the quasi-coherent module $\tilde{M}$. The proof for the exactness of this sequence is similar to the general case.

There is an alternative proof: Consider the commutative $A$-algebra $B=\prod_{i \in I} A[f_i^{-1}]$ and observe that it is faithfully flat. Thus, it suffices to prove that $(*)$ tensored with $B$ is exact. However, it turns out that this tensored sequence is split exact. (Of course, one has to write down the splittings.)

Answer 2

Something worth mentioning: in the association $\underline{\operatorname{SmVar}}_\mathbb C \to \underline{\operatorname{Mfd}}_\mathbb C$, affine varieties correspond to Stein manifolds. For the latter, vanishing of the higher cohomology of a coherent sheaf is essentially the content of Cartan's theorem B.

The case of quasi-coherent sheaves on affine schemes can be seen as a massive generalisation of the algebraic analogue. Note however that for non-projective varieties $X$, one should not expect the natural map $H^i(X,\mathscr F) \to H^i(X^{\operatorname{an}},\mathscr F^{\operatorname{an}})$ to be an isomorphism (think about $H^0(\mathbb A^1, \mathcal O_{\mathbb A^1})$). Thus, Cartan's theorem B really proves a different statement. In fact, it is an important ingredient in Serre's proof of the GAGA principle for projective varieties.